Despite the importance of continental breakup in plate tectonics, precisely how extensional processes such as brittle faulting, ductile plate stretching, and magma intrusion evolve in space and time during the development of new ocean basins remains poorly understood. The rifting of Arabia from Africa in the Afar depression is an ideal natural laboratory to address this problem since the region exposes subaerially the tectonically active transition from continental rifting to incipient seafloor spreading. We review recent constraints on along-axis variations in rift morphology, crustal and mantle structure, the distribution and style of ongoing faulting, subsurface magmatism and surface volcanism in the Red Sea rift of Afar to understand processes ultimately responsible for the formation of magmatic rifted continental margins. Our synthesis shows that there is a fundamental change in rift morphology from central Afar northward into the Danakil depression, spatially coincident with marked thinning of the crust, an increase in the volume of young basalt flows, and subsidence of the land towards and below sea-level. The variations can be attributed to a northward increase in proportion of extension by ductile plate stretching at the expense of magma intrusion. This is likely in response to a longer history of localised heating and weakening in a narrower rift. Thus, although magma intrusion accommodates strain for a protracted period during rift development, the final stages of breakup are dominated by a phase of plate stretching with a shift from intrusive to extrusive magmatism. This late-stage pulse of decompression melting due to plate thinning may be responsible for the formation of seaward dipping reflector sequences of basalts and sediments, which are ubiquitous at magmatic rifted margins worldwide.